On L1L^1-estimates of derivatives of univalent rational functions

We study the growth of the quantity $\int_{\mathbb{T}}|R'(z)|\,dm(z)$ for rational functions $R$ of degree $n$, which are bounded and univalent in the unit disk, and prove that this quantity may grow as $n^\gamma$, $\gamma>0$, when $n\to\infty$. Some applications of this result to problems of regularity of boundaries of Nevanlinna domains are considered. We also discuss a related result by Dolzhenko which applies to general (non-univalent) rational functions.Comment: 16 pages, to appear in Journal d'Analyse Mathematiqu

Interaction of the Laws of Electrodynamics in the Huber Effect

A complex physical phenomenon, first discovered by engineer J. Huber in 1951, is investigated. From the perspective of an external observer, the phenomenon is as follows: an electric current is passed through the wheel pairs of the car moving from the rail to the rail. The current, passing through the movable contacts of the wheels and rails, creates an additional (up to the moment of inertia) torque. The research task is to explain the reason for the occurrence of torque. Based on the analysis of individual components of the electrodynamic phenomenon discovered by Huber, an algorithm for the successive interaction of the individual components of the effect is found on the basis of the laws of classical electrodynamics: electric, ferromagnetic, and mechanical.The identity of the effect is explained, both for the wheel pair and for the bearing (Kosyrev-Milroy engine). For the first time, the cause of the appearance of the torque is revealed: relative movement of surface charges in the region of the movable electrical contact to the wheel body and the rails (or balls and guides). Moving charges unevenly magnetized ferromagnetic bodies according to the Biot-Savart-Laplace law. Due to the reduction in the clearance of the oncoming side of the wheel (or balls) and the increase on the trailing side, the pulling force from the oncoming side and, accordingly, the moment are more than on trailing side. The presented theoretical explanations completely correspond to the experimental investigation of the effect carried out by different scientists at different times

A mixed effects model for longitudinal relational and network data, with applications to international trade and conflict

The focus of this paper is an approach to the modeling of longitudinal social network or relational data. Such data arise from measurements on pairs of objects or actors made at regular temporal intervals, resulting in a social network for each point in time. In this article we represent the network and temporal dependencies with a random effects model, resulting in a stochastic process defined by a set of stationary covariance matrices. Our approach builds upon the social relations models of Warner, Kenny and Stoto [Journal of Personality and Social Psychology 37 (1979) 1742--1757] and Gill and Swartz [Canad. J. Statist. 29 (2001) 321--331] and allows for an intra- and inter-temporal representation of network structures. We apply the methodology to two longitudinal data sets: international trade (continuous response) and militarized interstate disputes (binary response).Comment: Published in at http://dx.doi.org/10.1214/10-AOAS403 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stochastic symmetry-breaking in a gaussian Hopfield model

We study a ``two-pattern'' Hopfield model with Gaussian disorder. We find that there are infinitely many pure states at low temperatures in this model, and we find that the metastate is supported on an infinity of symmetric pairs of pure states. The origin of this phenomenon is the random breaking of a rotation symmetry of the distribution of the disorderComment: 31pp, AMSTe